direct product, cyclic, abelian, monomial
Aliases: C30, also denoted Z30, SmallGroup(30,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C30 |
| C1 — C30 |
| C1 — C30 |
Generators and relations for C30
G = < a | a30=1 >
Character table of C30
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | -1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | ζ52 | ζ53 | ζ54 | ζ5 | 1 | 1 | ζ5 | ζ54 | ζ52 | ζ53 | ζ54 | ζ52 | ζ53 | ζ54 | ζ5 | ζ5 | ζ52 | ζ53 | ζ54 | ζ53 | ζ54 | ζ5 | ζ5 | ζ52 | ζ53 | ζ52 | linear of order 5 |
| ρ8 | 1 | -1 | 1 | 1 | ζ52 | ζ53 | ζ54 | ζ5 | -1 | -1 | -ζ5 | -ζ54 | -ζ52 | -ζ53 | ζ54 | ζ52 | ζ53 | ζ54 | ζ5 | ζ5 | ζ52 | ζ53 | -ζ54 | -ζ53 | -ζ54 | -ζ5 | -ζ5 | -ζ52 | -ζ53 | -ζ52 | linear of order 10 |
| ρ9 | 1 | 1 | ζ32 | ζ3 | ζ52 | ζ53 | ζ54 | ζ5 | ζ32 | ζ3 | ζ5 | ζ54 | ζ52 | ζ53 | ζ32ζ54 | ζ3ζ52 | ζ3ζ53 | ζ3ζ54 | ζ3ζ5 | ζ32ζ5 | ζ32ζ52 | ζ32ζ53 | ζ32ζ54 | ζ3ζ53 | ζ3ζ54 | ζ3ζ5 | ζ32ζ5 | ζ32ζ52 | ζ32ζ53 | ζ3ζ52 | linear of order 15 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | ζ52 | ζ53 | ζ54 | ζ5 | ζ6 | ζ65 | -ζ5 | -ζ54 | -ζ52 | -ζ53 | ζ32ζ54 | ζ3ζ52 | ζ3ζ53 | ζ3ζ54 | ζ3ζ5 | ζ32ζ5 | ζ32ζ52 | ζ32ζ53 | -ζ32ζ54 | -ζ3ζ53 | -ζ3ζ54 | -ζ3ζ5 | -ζ32ζ5 | -ζ32ζ52 | -ζ32ζ53 | -ζ3ζ52 | linear of order 30 faithful |
| ρ11 | 1 | 1 | ζ3 | ζ32 | ζ52 | ζ53 | ζ54 | ζ5 | ζ3 | ζ32 | ζ5 | ζ54 | ζ52 | ζ53 | ζ3ζ54 | ζ32ζ52 | ζ32ζ53 | ζ32ζ54 | ζ32ζ5 | ζ3ζ5 | ζ3ζ52 | ζ3ζ53 | ζ3ζ54 | ζ32ζ53 | ζ32ζ54 | ζ32ζ5 | ζ3ζ5 | ζ3ζ52 | ζ3ζ53 | ζ32ζ52 | linear of order 15 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | ζ52 | ζ53 | ζ54 | ζ5 | ζ65 | ζ6 | -ζ5 | -ζ54 | -ζ52 | -ζ53 | ζ3ζ54 | ζ32ζ52 | ζ32ζ53 | ζ32ζ54 | ζ32ζ5 | ζ3ζ5 | ζ3ζ52 | ζ3ζ53 | -ζ3ζ54 | -ζ32ζ53 | -ζ32ζ54 | -ζ32ζ5 | -ζ3ζ5 | -ζ3ζ52 | -ζ3ζ53 | -ζ32ζ52 | linear of order 30 faithful |
| ρ13 | 1 | 1 | 1 | 1 | ζ54 | ζ5 | ζ53 | ζ52 | 1 | 1 | ζ52 | ζ53 | ζ54 | ζ5 | ζ53 | ζ54 | ζ5 | ζ53 | ζ52 | ζ52 | ζ54 | ζ5 | ζ53 | ζ5 | ζ53 | ζ52 | ζ52 | ζ54 | ζ5 | ζ54 | linear of order 5 |
| ρ14 | 1 | -1 | 1 | 1 | ζ54 | ζ5 | ζ53 | ζ52 | -1 | -1 | -ζ52 | -ζ53 | -ζ54 | -ζ5 | ζ53 | ζ54 | ζ5 | ζ53 | ζ52 | ζ52 | ζ54 | ζ5 | -ζ53 | -ζ5 | -ζ53 | -ζ52 | -ζ52 | -ζ54 | -ζ5 | -ζ54 | linear of order 10 |
| ρ15 | 1 | 1 | ζ32 | ζ3 | ζ54 | ζ5 | ζ53 | ζ52 | ζ32 | ζ3 | ζ52 | ζ53 | ζ54 | ζ5 | ζ32ζ53 | ζ3ζ54 | ζ3ζ5 | ζ3ζ53 | ζ3ζ52 | ζ32ζ52 | ζ32ζ54 | ζ32ζ5 | ζ32ζ53 | ζ3ζ5 | ζ3ζ53 | ζ3ζ52 | ζ32ζ52 | ζ32ζ54 | ζ32ζ5 | ζ3ζ54 | linear of order 15 |
| ρ16 | 1 | -1 | ζ32 | ζ3 | ζ54 | ζ5 | ζ53 | ζ52 | ζ6 | ζ65 | -ζ52 | -ζ53 | -ζ54 | -ζ5 | ζ32ζ53 | ζ3ζ54 | ζ3ζ5 | ζ3ζ53 | ζ3ζ52 | ζ32ζ52 | ζ32ζ54 | ζ32ζ5 | -ζ32ζ53 | -ζ3ζ5 | -ζ3ζ53 | -ζ3ζ52 | -ζ32ζ52 | -ζ32ζ54 | -ζ32ζ5 | -ζ3ζ54 | linear of order 30 faithful |
| ρ17 | 1 | 1 | ζ3 | ζ32 | ζ54 | ζ5 | ζ53 | ζ52 | ζ3 | ζ32 | ζ52 | ζ53 | ζ54 | ζ5 | ζ3ζ53 | ζ32ζ54 | ζ32ζ5 | ζ32ζ53 | ζ32ζ52 | ζ3ζ52 | ζ3ζ54 | ζ3ζ5 | ζ3ζ53 | ζ32ζ5 | ζ32ζ53 | ζ32ζ52 | ζ3ζ52 | ζ3ζ54 | ζ3ζ5 | ζ32ζ54 | linear of order 15 |
| ρ18 | 1 | -1 | ζ3 | ζ32 | ζ54 | ζ5 | ζ53 | ζ52 | ζ65 | ζ6 | -ζ52 | -ζ53 | -ζ54 | -ζ5 | ζ3ζ53 | ζ32ζ54 | ζ32ζ5 | ζ32ζ53 | ζ32ζ52 | ζ3ζ52 | ζ3ζ54 | ζ3ζ5 | -ζ3ζ53 | -ζ32ζ5 | -ζ32ζ53 | -ζ32ζ52 | -ζ3ζ52 | -ζ3ζ54 | -ζ3ζ5 | -ζ32ζ54 | linear of order 30 faithful |
| ρ19 | 1 | 1 | 1 | 1 | ζ5 | ζ54 | ζ52 | ζ53 | 1 | 1 | ζ53 | ζ52 | ζ5 | ζ54 | ζ52 | ζ5 | ζ54 | ζ52 | ζ53 | ζ53 | ζ5 | ζ54 | ζ52 | ζ54 | ζ52 | ζ53 | ζ53 | ζ5 | ζ54 | ζ5 | linear of order 5 |
| ρ20 | 1 | -1 | 1 | 1 | ζ5 | ζ54 | ζ52 | ζ53 | -1 | -1 | -ζ53 | -ζ52 | -ζ5 | -ζ54 | ζ52 | ζ5 | ζ54 | ζ52 | ζ53 | ζ53 | ζ5 | ζ54 | -ζ52 | -ζ54 | -ζ52 | -ζ53 | -ζ53 | -ζ5 | -ζ54 | -ζ5 | linear of order 10 |
| ρ21 | 1 | 1 | ζ32 | ζ3 | ζ5 | ζ54 | ζ52 | ζ53 | ζ32 | ζ3 | ζ53 | ζ52 | ζ5 | ζ54 | ζ32ζ52 | ζ3ζ5 | ζ3ζ54 | ζ3ζ52 | ζ3ζ53 | ζ32ζ53 | ζ32ζ5 | ζ32ζ54 | ζ32ζ52 | ζ3ζ54 | ζ3ζ52 | ζ3ζ53 | ζ32ζ53 | ζ32ζ5 | ζ32ζ54 | ζ3ζ5 | linear of order 15 |
| ρ22 | 1 | -1 | ζ32 | ζ3 | ζ5 | ζ54 | ζ52 | ζ53 | ζ6 | ζ65 | -ζ53 | -ζ52 | -ζ5 | -ζ54 | ζ32ζ52 | ζ3ζ5 | ζ3ζ54 | ζ3ζ52 | ζ3ζ53 | ζ32ζ53 | ζ32ζ5 | ζ32ζ54 | -ζ32ζ52 | -ζ3ζ54 | -ζ3ζ52 | -ζ3ζ53 | -ζ32ζ53 | -ζ32ζ5 | -ζ32ζ54 | -ζ3ζ5 | linear of order 30 faithful |
| ρ23 | 1 | 1 | ζ3 | ζ32 | ζ5 | ζ54 | ζ52 | ζ53 | ζ3 | ζ32 | ζ53 | ζ52 | ζ5 | ζ54 | ζ3ζ52 | ζ32ζ5 | ζ32ζ54 | ζ32ζ52 | ζ32ζ53 | ζ3ζ53 | ζ3ζ5 | ζ3ζ54 | ζ3ζ52 | ζ32ζ54 | ζ32ζ52 | ζ32ζ53 | ζ3ζ53 | ζ3ζ5 | ζ3ζ54 | ζ32ζ5 | linear of order 15 |
| ρ24 | 1 | -1 | ζ3 | ζ32 | ζ5 | ζ54 | ζ52 | ζ53 | ζ65 | ζ6 | -ζ53 | -ζ52 | -ζ5 | -ζ54 | ζ3ζ52 | ζ32ζ5 | ζ32ζ54 | ζ32ζ52 | ζ32ζ53 | ζ3ζ53 | ζ3ζ5 | ζ3ζ54 | -ζ3ζ52 | -ζ32ζ54 | -ζ32ζ52 | -ζ32ζ53 | -ζ3ζ53 | -ζ3ζ5 | -ζ3ζ54 | -ζ32ζ5 | linear of order 30 faithful |
| ρ25 | 1 | 1 | 1 | 1 | ζ53 | ζ52 | ζ5 | ζ54 | 1 | 1 | ζ54 | ζ5 | ζ53 | ζ52 | ζ5 | ζ53 | ζ52 | ζ5 | ζ54 | ζ54 | ζ53 | ζ52 | ζ5 | ζ52 | ζ5 | ζ54 | ζ54 | ζ53 | ζ52 | ζ53 | linear of order 5 |
| ρ26 | 1 | -1 | 1 | 1 | ζ53 | ζ52 | ζ5 | ζ54 | -1 | -1 | -ζ54 | -ζ5 | -ζ53 | -ζ52 | ζ5 | ζ53 | ζ52 | ζ5 | ζ54 | ζ54 | ζ53 | ζ52 | -ζ5 | -ζ52 | -ζ5 | -ζ54 | -ζ54 | -ζ53 | -ζ52 | -ζ53 | linear of order 10 |
| ρ27 | 1 | 1 | ζ32 | ζ3 | ζ53 | ζ52 | ζ5 | ζ54 | ζ32 | ζ3 | ζ54 | ζ5 | ζ53 | ζ52 | ζ32ζ5 | ζ3ζ53 | ζ3ζ52 | ζ3ζ5 | ζ3ζ54 | ζ32ζ54 | ζ32ζ53 | ζ32ζ52 | ζ32ζ5 | ζ3ζ52 | ζ3ζ5 | ζ3ζ54 | ζ32ζ54 | ζ32ζ53 | ζ32ζ52 | ζ3ζ53 | linear of order 15 |
| ρ28 | 1 | -1 | ζ32 | ζ3 | ζ53 | ζ52 | ζ5 | ζ54 | ζ6 | ζ65 | -ζ54 | -ζ5 | -ζ53 | -ζ52 | ζ32ζ5 | ζ3ζ53 | ζ3ζ52 | ζ3ζ5 | ζ3ζ54 | ζ32ζ54 | ζ32ζ53 | ζ32ζ52 | -ζ32ζ5 | -ζ3ζ52 | -ζ3ζ5 | -ζ3ζ54 | -ζ32ζ54 | -ζ32ζ53 | -ζ32ζ52 | -ζ3ζ53 | linear of order 30 faithful |
| ρ29 | 1 | 1 | ζ3 | ζ32 | ζ53 | ζ52 | ζ5 | ζ54 | ζ3 | ζ32 | ζ54 | ζ5 | ζ53 | ζ52 | ζ3ζ5 | ζ32ζ53 | ζ32ζ52 | ζ32ζ5 | ζ32ζ54 | ζ3ζ54 | ζ3ζ53 | ζ3ζ52 | ζ3ζ5 | ζ32ζ52 | ζ32ζ5 | ζ32ζ54 | ζ3ζ54 | ζ3ζ53 | ζ3ζ52 | ζ32ζ53 | linear of order 15 |
| ρ30 | 1 | -1 | ζ3 | ζ32 | ζ53 | ζ52 | ζ5 | ζ54 | ζ65 | ζ6 | -ζ54 | -ζ5 | -ζ53 | -ζ52 | ζ3ζ5 | ζ32ζ53 | ζ32ζ52 | ζ32ζ5 | ζ32ζ54 | ζ3ζ54 | ζ3ζ53 | ζ3ζ52 | -ζ3ζ5 | -ζ32ζ52 | -ζ32ζ5 | -ζ32ζ54 | -ζ3ζ54 | -ζ3ζ53 | -ζ3ζ52 | -ζ32ζ53 | linear of order 30 faithful |
►Regular action on 30 points - transitive group 30T1
Generators in S30
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30)
G:=sub<Sym(30)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)]])
G:=TransitiveGroup(30,1);
C30 is a maximal subgroup of
Dic15
| action | f(x) | Disc(f) |
|---|---|---|
| 30T1 | x30+x29+x28+x27+x26+x25+x24+x23+x22+x21+x20+x19+x18+x17+x16+x15+x14+x13+x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1 | -3129 |
Matrix representation of C30 ►in GL2(𝔽11) generated by
| 0 | 6 |
| 4 | 3 |
G:=sub<GL(2,GF(11))| [0,4,6,3] >;
C30 in GAP, Magma, Sage, TeX
C_{30} % in TeX
G:=Group("C30"); // GroupNames label
G:=SmallGroup(30,4);
// by ID
G=gap.SmallGroup(30,4);
# by ID
G:=PCGroup([3,-2,-3,-5]);
// Polycyclic
G:=Group<a|a^30=1>;
// generators/relations
Export
Subgroup lattice of C30 in TeX
Character table of C30 in TeX